A right prism has a triangular base whose sides are 13 cm, 20 cm and 21 cm. If the altitude of the prism is 9 cm, then its volume is

Options

A. a) `1314 cm^(3)`
B. b) `1134 cm^(3)`
C. c) `1413 cm^(3)`
D. d) `1143 cm^(3)`

Answer

To find the volume of a right prism with a triangular base, we need to follow these steps:

### Step 1: Identify the sides of the triangular base
The sides of the triangular base are given as:
- \( a = 13 \, \text{cm} \)
- \( b = 20 \, \text{cm} \)
- \( c = 21 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter of the triangle
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{13 + 20 + 21}{2} = \frac{54}{2} = 27 \, \text{cm} \]

### Step 3: Calculate the area of the triangle using Heron's formula
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ A = \sqrt{27(27-13)(27-20)(27-21)} \]
Calculating each term:
- \( s - a = 27 - 13 = 14 \)
- \( s - b = 27 - 20 = 7 \)
- \( s - c = 27 - 21 = 6 \)

Now substituting these values back into the area formula:
\[ A = \sqrt{27 \times 14 \times 7 \times 6} \]

### Step 4: Calculate the product inside the square root
Calculating the product:
\[ 27 \times 14 = 378 \]
\[ 378 \times 7 = 2646 \]
\[ 2646 \times 6 = 15876 \]
Now, taking the square root:
\[ A = \sqrt{15876} \approx 126 \, \text{cm}^2 \]

### Step 5: Calculate the volume of the prism
The volume \( V \) of the prism is given by the formula:
\[ V = \text{Area of base} \times \text{Height} \]
Given that the height \( h = 9 \, \text{cm} \):
\[ V = 126 \times 9 = 1134 \, \text{cm}^3 \]

### Final Answer
The volume of the prism is \( 1134 \, \text{cm}^3 \).

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